I'm afraid I can't turn everyone into demon arcade PCB 'fixers'! Its not that it's amazingly complicated or anything (although I do still own some boards which have completely foxed me of how to be fixed) I think of it as more of a puzzle that you're trying to solve and I'm afraid that some people just don't seem to 'get it'!!! I will however try my very best to explain it simply, but bear in mind this is just a guide - it's not an instruction manual. There's going to be a whole lot of using your own initiative going on as well...
The first 'step' in the puzzle solving is to establish exactly what is wrong...

• What's wrong?

It may sound obvious to start with "what's wrong?" but it's blatantly of great importance. Say you've a machine which has arrived DOA, but you can hear the game playing when it's switched on. On most occasions you'd be quite right to assume the monitor is dead, but does it really hurt to test it? How much will you be kicking yourself after you've gone to the trouble of getting all your equipment out and building a loom if it just turns out to be that the monitor input molex plug just became loose in transit and no signal is being received by the monitor. There's a few things like this which occur time and time again when moving machines, always check the obvious when maybe, the sound stops working (maybe the crimp connector has fallen off the speaker) or that an input suddenly isn't working (again, maybe the crimp connector is laying in the bottom of the control panel not connected to anything). I cannot overstate this enough, NEVER overlook the obvious!!!
Now assuming we have a board fault and not a machine fault we can move on (I'm not going to cover fixing monitors as to be fair I've not the experience to be telling others how to handle such dangerous equipment. Randy Fromm has more than enough information on his 'secret technical server' anyway).
Here's a little probably incomplete list of board related 'obvious' things!

1. Is the board receiving the right voltages? Try adjusting the +5v up to +5.1v - this has solved a lot of problems for me in the past. Bear in mind that the voltage is different when the board isn't connected. You have to adjust it while the board is live to get a correct reading. This is because the board is drawing on the voltage and 'using' it, therefore the voltage will be higher when the board isn't attached.
2. Is the loom lined up properly and are all the pins on the edge connector present? Another obvious one, but yet it does happen! I had an arkanoid cabaret that had a pin in the edge connector all squashed upwards. Fortunately all I had to do was bend it out again (it was salvageable) and the game worked fine. Another game I worked on (not one of mine) simply had an edge connect which was slightly too long for the pins it sat on! All that had happened was the pins had become misaligned with the board and simply pushing it slightly more in the middle made the game come to life!
3. Are all the chips seated properly? Again, a common fault with PCBs sent through the post. You can't really expect a 20+ year old delicate piece of equipment to be fine when it's wrapped just in newspaper and sent through the mail! Press down on all the socketed chips while the game is running. You may see some change on the screen, and you should hopefully be able to pinpoint which chip it is which is causing the fault. If any of the component leads are black, this can cause a lack of connectivity to the chips socket. Try cleaning the legs with a pencil eraser it should get most of it off. You can spray the socket with contact cleaner (and the legs too) to make a better connection. If all else fails, then perhaps remove and replace all the socketed chips one-by-one. The same goes for daughterboards - I bought a faulty Street Fighter 2 PCB and all it required was the daughterboard being reseated (the pins weren't aligned!) - Quickest fix I've ever had!!!
4. Are all the components attached fully? On the flipside of not wrapping it well enough, I've been guilty of over wrapping before and the PCB getting a bit squashed! Check that nothing is loose (especially on capacitors and things with long leads) also check that no long leads have been squashed together on the solder side and are shorting together. All these things have effect!
5. Are any ICs inserted the wrong way round?!!! Usually you can expect an IC to pop when it's inserted the wrong way round, but this isn't always the case. I once had a board arrive with all the chip markings reading the right way, but one PROM had the notch the wrong way round! The notch signifies the top of the chip (not the direction of the markings) I turned it around and unfortunately the board still didn't work! But still, I would have been kicking myself later on if I hadn't of noticed!
6. Are the chips in the correct sockets? Maybe not an obvious thing to be able to figure out! I bought a faulty Pacland PCB from eBay and all it required was two of the custom ICs being swapped over as they were in the wrong places!!!
7. Is anything missing? If you buy a PCB marked 'untested' from eBay then 99 times out of 100 that means 'broken'... Be careful you haven't been sold a donor board... Those chips are missing for a reason and the reason is possibly that they've no modern equivalent and will just sit on your shelf gathering dust :o( Note though that some (and quite a few) boards have sockets which are meant to be unpopulated. If you're concerned that you're missing something then try to find the schematics online or even perhaps make an appeal on a messageboard like UKVAC. Someone somewhere will know! :o)
8. Is anything too hot to touch! Components fail in a few different ways, and to me it's always quite a relief when one is too hot to touch! I fixed my Battlezone just by passing my hand lightly over the PCB and getting VERY burnt on one particular IC! I replaced it, and the game ran just fine again! :o) This doesn't always happen unfortunately, but it's a good place to start! Note here though that certain parts such as colour PROMs do actually run quite hot anyway. Don't be misled!
9. Is the problem with the loom? I had to travel over 400 miles before to visit a shop I'd sold a PCB to (which they said arrived dead) I'd given them a loom with it that I hadn't built (it came on a different PCB I'd bought from eBay and the boards shared pinouts). The game had worked fine on my set up, so I took my own loom with me just in case... The problem turned out to be that the game was fussy with missing ground connections! The loom I'd given them didn't have all the grounds hooked up - that coupled with the awful supergun they used in the shop (which also, didn't have all the grounds hooked up) meant the game wouldn't run! Fortunately, it was more tolerant with my own loom so I left it with them (I didn't receive any compensation for the 400 miles travelling though!)
10. Am I using the right loom? Can and does happen! A sudden lapse in concentration (especially if you have a few boards in front of you!) and you may well be plugging the wrong loom onto your game! Take care!!! Check the "Can I play games that aren't JAMMA?" for some pinouts. There's some hints on the "Loom Making Pictorial Guide" as well about how to recognise some obvious things by sight with regard to the power connections. Don't forget as well that 28 way does not always mean it's JAMMA!!!
11. Are there any broken traces? This isn't always very easy to see and sometimes traces get broken under sockets where someone has been too heavyhanded using a screwdriver to remove a chip! Still, large scratches are easily noticed and it's always best to check for continuity across them. Use proper patch wire to repair them; don't just think that a little solder will hold!
12. Has a repair already been attempted? A lot of boards I see have messy hacks and/ or failed repairs on them. I do tend to start by undo-ing other peoples work... Maybe that's not the best way to go about it - but then I'd rather know for myself that the repair isn't the cause of another fault. Especially with hacks... I've seen some *weird* things in my time, which have actually worked!!! But still, for peace of mind - I most of the time think it's best to find out for yourself if a repair or hack is 'good'. One note though, is not to confuse factory hacks with repair hacks... PCBs like Bombjack and Star Wars have resistors and/ or jumper wires which are supposed to be there! Never assume anything!

Most of these things are purely common sense and I've probably missed a few as well. That last line says an awful lot - "Never assume anything!" it's possibly the best lesson anyone can learn about anything! Keep it in mind!
Please just use a little imagination and a lot of common sense!!!

• Where is the fault?

Okay so this probably sounds like cheating :o) but yes, it is a new section! Arcade boards are lots of separate circuits which are interlinked so that the game can access them and use them. Most of this is covered by reading the "How do games work?" section - you can safely ignore the programming section if you only want to read about fixing PCBs. The point here is that it'll be no help to you if you're trying to figure out a problem in the sound section if you're prodding around in the watchdog circuit! It's also infeasible (usually) to test every single component on a PCB which you don't have schematics for. Knowing where to look on a PCB for the symptons it's showing is half the battle...
Being simplistic I guess you can break an arcade board down into three separate areas - CPU, graphics and sound. Of course, these sections in themselves break down even further - but this is only a guide, I couldn't possibly cover eventualities for every single problem you might come across for every single PCB... For a start, I've still got boards that I can't fix!